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2006 Fractional heat equations on a segment
Elena I. Kaikina
Differential Integral Equations 19(8): 891-918 (2006).

Abstract

We study the initial-boundary-value problem for the fractional heat equation on a segment $ ( 0,a ) $ \begin{equation} \left \{ \begin{array}{c} u_{t}+\lambda \vert u \vert ^{\rho }\text{ }u+C_{\alpha }\partial _{x}^{\alpha }u=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right . \tag*{(0.1)} \end{equation} where $\lambda \in R$ , $\rho \geq 0$ , $\alpha \in ( 1,\frac{3}{2} ] $ $,$ the constant $C_{\alpha }$ is chosen by a dissipative condition, such that $\ \text{Re}C_{\alpha }p^{ [ \alpha ] +1- \{ \alpha \} }>0$ for $\text{Re}p=0$ and \begin{equation*} \partial _{x}^{\alpha }u=\int_{0}^{x}\frac{\partial _{s}^{ [ \alpha ] +1}u(s,t)}{ ( x-s ) ^{ \{ \alpha \} }}ds. \end{equation*} Here $ [ \alpha ] $ and $ \{ \alpha \} $ are integer and fractional parts of $\alpha .$ The aim of this paper is to prove the global existence of solutions to the initial-boundary-value problem (0.1) and to find the main term of the asymptotic representation of solutions.

Citation

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Elena I. Kaikina. "Fractional heat equations on a segment." Differential Integral Equations 19 (8) 891 - 918, 2006.

Information

Published: 2006
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35214
MathSciNet: MR2263434

Subjects:
Primary: 35K55
Secondary: 26A33 , 35C20

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.19 • No. 8 • 2006
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